update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_1 / pr3 / subst1.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma
index 048fc9bebf363085918831e8dd13ddf80186f85a..3c2e56c05b03c614a9ed4b58d25308d5594814fd 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma
@@ -18,7 +18,7 @@ include "basic_1/pr3/fwd.ma".
 
 include "basic_1/pr2/subst1.ma".
 
-theorem pr3_subst1:
+lemma pr3_subst1:
  \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c 
 (CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) 
 \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr3 c 
@@ -26,35 +26,26 @@ w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
 \def
  \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda 
 (H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (H0: (pr3 c t1 t2)).(let TMP_3 \def (\lambda (t: T).(\lambda (t0: 
-T).(\forall (w1: T).((subst1 i v t w1) \to (let TMP_1 \def (\lambda (w2: 
-T).(pr3 c w1 w2)) in (let TMP_2 \def (\lambda (w2: T).(subst1 i v t0 w2)) in 
-(ex2 T TMP_1 TMP_2))))))) in (let TMP_7 \def (\lambda (t: T).(\lambda (w1: 
-T).(\lambda (H1: (subst1 i v t w1)).(let TMP_4 \def (\lambda (w2: T).(pr3 c 
-w1 w2)) in (let TMP_5 \def (\lambda (w2: T).(subst1 i v t w2)) in (let TMP_6 
-\def (pr3_refl c w1) in (ex_intro2 T TMP_4 TMP_5 w1 TMP_6 H1))))))) in (let 
-TMP_25 \def (\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 
-t3)).(\lambda (t5: T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall 
-(w1: T).((subst1 i v t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) 
-(\lambda (w2: T).(subst1 i v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: 
-(subst1 i v t4 w1)).(let TMP_8 \def (\lambda (w2: T).(pr2 c w1 w2)) in (let 
-TMP_9 \def (\lambda (w2: T).(subst1 i v t3 w2)) in (let TMP_10 \def (\lambda 
-(w2: T).(pr3 c w1 w2)) in (let TMP_11 \def (\lambda (w2: T).(subst1 i v t5 
-w2)) in (let TMP_12 \def (ex2 T TMP_10 TMP_11) in (let TMP_23 \def (\lambda 
-(x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3 x)).(let 
-TMP_13 \def (\lambda (w2: T).(pr3 c x w2)) in (let TMP_14 \def (\lambda (w2: 
-T).(subst1 i v t5 w2)) in (let TMP_15 \def (\lambda (w2: T).(pr3 c w1 w2)) in 
-(let TMP_16 \def (\lambda (w2: T).(subst1 i v t5 w2)) in (let TMP_17 \def 
-(ex2 T TMP_15 TMP_16) in (let TMP_21 \def (\lambda (x0: T).(\lambda (H7: (pr3 
-c x x0)).(\lambda (H8: (subst1 i v t5 x0)).(let TMP_18 \def (\lambda (w2: 
-T).(pr3 c w1 w2)) in (let TMP_19 \def (\lambda (w2: T).(subst1 i v t5 w2)) in 
-(let TMP_20 \def (pr3_sing c x w1 H5 x0 H7) in (ex_intro2 T TMP_18 TMP_19 x0 
-TMP_20 H8))))))) in (let TMP_22 \def (H3 x H6) in (ex2_ind T TMP_13 TMP_14 
-TMP_17 TMP_21 TMP_22))))))))))) in (let TMP_24 \def (pr2_subst1 c e v i H t4 
-t3 H1 w1 H4) in (ex2_ind T TMP_8 TMP_9 TMP_12 TMP_23 TMP_24)))))))))))))))) 
-in (pr3_ind c TMP_3 TMP_7 TMP_25 t1 t2 H0))))))))))).
+T).(\lambda (H0: (pr3 c t1 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: 
+T).(\forall (w1: T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 
+w2)) (\lambda (w2: T).(subst1 i v t0 w2))))))) (\lambda (t: T).(\lambda (w1: 
+T).(\lambda (H1: (subst1 i v t w1)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 
+w2)) (\lambda (w2: T).(subst1 i v t w2)) w1 (pr3_refl c w1) H1)))) (\lambda 
+(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 t3)).(\lambda (t5: 
+T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall (w1: T).((subst1 i v 
+t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i 
+v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: (subst1 i v t4 w1)).(ex2_ind T 
+(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T 
+(\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 w2))) 
+(\lambda (x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3 
+x)).(ex2_ind T (\lambda (w2: T).(pr3 c x w2)) (\lambda (w2: T).(subst1 i v t5 
+w2)) (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 
+w2))) (\lambda (x0: T).(\lambda (H7: (pr3 c x x0)).(\lambda (H8: (subst1 i v 
+t5 x0)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 
+i v t5 w2)) x0 (pr3_sing c x w1 H5 x0 H7) H8)))) (H3 x H6))))) (pr2_subst1 c 
+e v i H t4 t3 H1 w1 H4)))))))))) t1 t2 H0)))))))).
 
-theorem pr3_gen_cabbr:
+lemma pr3_gen_cabbr:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
 (e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) 
 \to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d 
@@ -63,50 +54,36 @@ a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T
 x1 x2))))))))))))))))
 \def
  \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
-t2)).(let TMP_5 \def (\lambda (t: T).(\lambda (t0: T).(\forall (e: 
-C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to 
-(\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) 
-\to (\forall (x1: T).((subst1 d u t (lift (S O) d x1)) \to (let TMP_3 \def 
-(\lambda (x2: T).(let TMP_1 \def (S O) in (let TMP_2 \def (lift TMP_1 d x2) 
-in (subst1 d u t0 TMP_2)))) in (let TMP_4 \def (\lambda (x2: T).(pr3 a x1 
-x2)) in (ex2 T TMP_3 TMP_4))))))))))))))) in (let TMP_11 \def (\lambda (t: 
-T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (_: (getl d c 
-(CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_: (csubst1 d u c 
-a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0 a)).(\lambda (x1: 
-T).(\lambda (H3: (subst1 d u t (lift (S O) d x1))).(let TMP_8 \def (\lambda 
-(x2: T).(let TMP_6 \def (S O) in (let TMP_7 \def (lift TMP_6 d x2) in (subst1 
-d u t TMP_7)))) in (let TMP_9 \def (\lambda (x2: T).(pr3 a x1 x2)) in (let 
-TMP_10 \def (pr3_refl a x1) in (ex_intro2 T TMP_8 TMP_9 x1 H3 
-TMP_10))))))))))))))) in (let TMP_39 \def (\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 
-t4)).(\lambda (H2: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl 
-d c (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to 
-(\forall (a: C).((drop (S O) d a0 a) \to (\forall (x1: T).((subst1 d u t0 
-(lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d 
-x2))) (\lambda (x2: T).(pr3 a x1 x2))))))))))))))).(\lambda (e: C).(\lambda 
-(u: T).(\lambda (d: nat).(\lambda (H3: (getl d c (CHead e (Bind Abbr) 
-u))).(\lambda (a0: C).(\lambda (H4: (csubst1 d u c a0)).(\lambda (a: 
-C).(\lambda (H5: (drop (S O) d a0 a)).(\lambda (x1: T).(\lambda (H6: (subst1 
-d u t3 (lift (S O) d x1))).(let TMP_14 \def (\lambda (x2: T).(let TMP_12 \def 
-(S O) in (let TMP_13 \def (lift TMP_12 d x2) in (subst1 d u t0 TMP_13)))) in 
-(let TMP_15 \def (\lambda (x2: T).(pr2 a x1 x2)) in (let TMP_18 \def (\lambda 
-(x2: T).(let TMP_16 \def (S O) in (let TMP_17 \def (lift TMP_16 d x2) in 
-(subst1 d u t4 TMP_17)))) in (let TMP_19 \def (\lambda (x2: T).(pr3 a x1 x2)) 
-in (let TMP_20 \def (ex2 T TMP_18 TMP_19) in (let TMP_37 \def (\lambda (x: 
-T).(\lambda (H7: (subst1 d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 
-x)).(let TMP_23 \def (\lambda (x2: T).(let TMP_21 \def (S O) in (let TMP_22 
-\def (lift TMP_21 d x2) in (subst1 d u t4 TMP_22)))) in (let TMP_24 \def 
-(\lambda (x2: T).(pr3 a x x2)) in (let TMP_27 \def (\lambda (x2: T).(let 
-TMP_25 \def (S O) in (let TMP_26 \def (lift TMP_25 d x2) in (subst1 d u t4 
-TMP_26)))) in (let TMP_28 \def (\lambda (x2: T).(pr3 a x1 x2)) in (let TMP_29 
-\def (ex2 T TMP_27 TMP_28) in (let TMP_35 \def (\lambda (x0: T).(\lambda (H9: 
-(subst1 d u t4 (lift (S O) d x0))).(\lambda (H10: (pr3 a x x0)).(let TMP_32 
-\def (\lambda (x2: T).(let TMP_30 \def (S O) in (let TMP_31 \def (lift TMP_30 
-d x2) in (subst1 d u t4 TMP_31)))) in (let TMP_33 \def (\lambda (x2: T).(pr3 
-a x1 x2)) in (let TMP_34 \def (pr3_sing a x x1 H8 x0 H10) in (ex_intro2 T 
-TMP_32 TMP_33 x0 H9 TMP_34))))))) in (let TMP_36 \def (H2 e u d H3 a0 H4 a H5 
-x H7) in (ex2_ind T TMP_23 TMP_24 TMP_29 TMP_35 TMP_36))))))))))) in (let 
-TMP_38 \def (pr2_gen_cabbr c t3 t0 H0 e u d H3 a0 H4 a H5 x1 H6) in (ex2_ind 
-T TMP_14 TMP_15 TMP_20 TMP_37 TMP_38)))))))))))))))))))))))) in (pr3_ind c 
-TMP_5 TMP_11 TMP_39 t1 t2 H))))))).
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (e: C).(\forall (u: 
+T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: 
+C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall 
+(x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 
+d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2))))))))))))))) 
+(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda 
+(_: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_: 
+(csubst1 d u c a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0 
+a)).(\lambda (x1: T).(\lambda (H3: (subst1 d u t (lift (S O) d 
+x1))).(ex_intro2 T (\lambda (x2: T).(subst1 d u t (lift (S O) d x2))) 
+(\lambda (x2: T).(pr3 a x1 x2)) x1 H3 (pr3_refl a x1))))))))))))) (\lambda 
+(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: 
+T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall (e: C).(\forall (u: 
+T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: 
+C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall 
+(x1: T).((subst1 d u t0 (lift (S O) d x1)) \to (ex2 T (\lambda (x2: 
+T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 
+x2))))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda 
+(H3: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H4: 
+(csubst1 d u c a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d a0 
+a)).(\lambda (x1: T).(\lambda (H6: (subst1 d u t3 (lift (S O) d 
+x1))).(ex2_ind T (\lambda (x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda 
+(x2: T).(pr2 a x1 x2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d 
+x2))) (\lambda (x2: T).(pr3 a x1 x2))) (\lambda (x: T).(\lambda (H7: (subst1 
+d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 x)).(ex2_ind T (\lambda 
+(x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x x2)) 
+(ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: 
+T).(pr3 a x1 x2))) (\lambda (x0: T).(\lambda (H9: (subst1 d u t4 (lift (S O) 
+d x0))).(\lambda (H10: (pr3 a x x0)).(ex_intro2 T (\lambda (x2: T).(subst1 d 
+u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)) x0 H9 (pr3_sing a x 
+x1 H8 x0 H10))))) (H2 e u d H3 a0 H4 a H5 x H7))))) (pr2_gen_cabbr c t3 t0 H0 
+e u d H3 a0 H4 a H5 x1 H6)))))))))))))))))) t1 t2 H)))).